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Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
Royal Institute of Technology.
2018 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed) Published
Abstract [en]

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

Place, publisher, year, edition, pages
Amsterdam: Elsevier, 2018. Vol. 330, p. 177-192
Keywords [en]
Nonlinear eigenvalue problems, Helmholtz problem, Scattering resonances, Dirichlet-to-Neumann map, Arnoldi's method, Matrix functions
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-138325DOI: 10.1016/j.cam.2017.08.012ISI: 000415783000014OAI: oai:DiVA.org:umu-138325DiVA, id: diva2:1134522
Available from: 2017-08-21 Created: 2017-08-21 Last updated: 2018-06-09Bibliographically approved

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Araujo-Cabarcas, Juan CarlosEngström, Christian

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